Some Technical Remarks on Negations of Discrete Probability Distributions and Their Information Loss

Negation of a discrete probability distribution was introduced by Yager. To date, several papers have been published discussing generalizations, properties, and applications negation. The recent work Wu et al. gives an excellent overview the literature motivation to deal with Our paper focuses on some technical aspects negation transformations. First, we prove that independent negations must be affine-linear. This fact established Batyrshin as open problem. Secondly, show repeated application leads progressive loss information (called monotonicity). In contrast literature, try obtain results not only for special but also general class ϕ-entropies. this framework, can need proven Yager transferred entire (=affine-linear) negations. For ϕ-entropies strictly concave generator function ϕ, increases separately sequences odd even numbers repetitions. By using Lagrangian approach, result extended, in neighbourhood uniform distribution, all repetition. Gini, Shannon, Havrda–Charvát (Tsallis), Rényi Sharma–Mittal entropy, has global minimum 0. dependent negations, it is easy analytical results. Therefore, simulate entropy how different affect Gini Shannon entropy. simulation approach advantage simplex vectors considered at once, rather than just arbitrarily selected vectors.